Ashya King was receiving treatment for a brain tumour at a Southampton hospital. Photograph: Hampshire police/PA

Police have defended the international search for a five-year-old boy with a brain tumour who was found in Spain after his parents removed him from a Southampton hospital without doctors' consent.

As the parents of Ashya King remained under arrest on the Costa del Sol, Hampshire police said it made "no apology" for the high-profile publicity campaign, which the distressed family said had cast them as "kidnappers" and turned them into refugees.

Brett King, 51, and his wife Naghemeh, 45, boarded a ferry from Portsmouth to Cherbourg with Ashya and his six siblings on Thursday. It sparked a Europe-wide hunt, with police warning that the boy's life was in danger unless he received urgent medical attention.

The couple were seeking medical treatment for Ashya in the Czech Republic that is unavailable on the NHS in the UK. They were arrested at a hotel in Vélez-Málaga after staff recognised the family and alerted Spanish police at 10pm on Saturday.

In a YouTube video shot before the arrest, King, a Jehovah's witness, cradled Ashya as he explained the family had decided to take him out of Southampton general hospital to seek a cancer treatment called proton beam, and because they feared doctors would seek an emergency protection order.

The case has sparked fierce debate, with Hampshire police being criticised for their handling of the appeal. As officers were en route to Spain to question the couple, who face extradition back to the UK, Asst Ch Const Chris Shead said the force had acted appropriately.

"I am aware that there has been a significant amount of debate going on around the appropriateness of police action in this case," he said.

Medical experts had told police the battery in Ashya's tube feeding unit was due to run down and his life was "in grave danger", he said.

"Faced with those circumstances I make no apology for the police being as proactive as they possibly can to find Ashya and ensure that he gets the help he needed. I would much sooner be standing here facing criticism for being proactive than to stand here and face criticism for doing nothing and eventually having to explain why a child has lost his life."

Ashya was taken to a high-dependency unit in Malaga on Saturday night, and moved to a lower-dependency unit on Sunday. A spokesman for Málaga's Hospital Materno Infantil said he was stable and that his life was not in danger.

His six siblings remained at the hotel where the family was found, police said.

The Spanish authorities were to decide whether Ashya's parents would be allowed to be with him on Sunday Extradition proceedings will now begin before a Spanish judge. The family was being supported by the Foreign and Commonwealth Office, said Shead.

A decision on whether the parents face prosecution would be taken by the Crown Prosecution Service.

In the YouTube video, Ashya's father said the family was "most disturbed" to see his face all over the internet and newspapers and that they had been "labelled as kidnappers putting his life at risk, neglect".

"As you can see there's nothing wrong with him, he is very happy actually since we took him out of hospital. He has been smiling a lot more, he has very much been interacting with us."

He said that surgeons "did a wonderful job" removing the tumour, but then his son went into "posterior fossa syndrome", with very limited moving or talking.

The family had requested proton beam treatment and were prepared to raise the money to pay for it. But King said his son's treatment seemed like "trial and error" and that he was told if he questioned the treatment the hospital would seek an emergency protection order.

The search for the family began in France, then switched to the Costa del Sol where the couple have family and business connections. Police already had a flat owned by the family near Estepona under surveillance and were tracing their phone calls in an effort to track them down.

Interpol had warned the police that they had 12 hours to find the child, as that was the length of the battery life of one of the machines that was keeping him alive.

It was when Danny King, 22, one of the couple's elder children, checked in to a hostel at La Esperanza de Benajarafe at 7.15pm on Saturday that the receptionist raised the alarm.

The parents were detained under a European detention order for alleged cruelty to a person under the age of 16 and were taken to the Vélez-Málaga police station. They will attend an extradition hearing presided over by judge Ismael Moreno at the central criminal court in Madrid on Monday.

King explained in the video that the family had been intending to seek proton beam treatment for Ashya in the Czech Republic.

"Proton beam is so much better for children with brain cancer," he said. "It zones in on the area, whereby normal radiation passes right through his head and comes out the other side and destroys everything in his head.

"We pleaded with them (in Southampton) for proton beam treatment. They looked at me straight in the face and said with his cancer – which is called medulloblastoma – it would have no benefit whatsoever.

"I went straight back to my room and looked it up and the American sites and French sites and Switzerland sites where they have proton beam said the opposite, it would be very beneficial for him."